Question

Consider Two Concentric circles C1: x2+y2-1=0 & C2 : x2+y2-4=0. A parabola is drawn through the points where ‘C1‘ meets the X-axis & having arbitrary tangent of ‘C2’ as its directrix. Then the locus of the focus of drawn parabola is ?????Please help me !! I am confused !!

Answer 1

Solution:

The two given concentric circles are,

$C_{1}: x^2+y^2-1=0 & C_{2} : x^2+y^2-4=0.$

having center (0,0) and radius 1 unit and 2 units respectively.

Also, it is given that, A parabola is drawn through the points where ‘$C_{1}$‘ meets the X-axis & having arbitrary tangent of  ‘$C_{2}$ as its Directrix.

As, the circle,  ‘$C_{1}$, cuts the X axis at, (1,0) and (-1,0) respectively.

We get this value by putting , y=0 in  ‘$C_{1}$.

$x^2=1\\\\ x=\pm1$

As, on value is negative and other is positive, there will be two parabolas passing through $(\pm1,0)$

Let the equation of parabola whose vertices is (h,k) is

→ $(x-h)^2= \pm4 a (y-k)$

→(1-h)²= 4 a (0-k)

→ (1-h)²= - 4 a k -----(1)

Also, (-1 -h)²= 4 a (0-k)

→ (1+h)²= - 4 a k  ------(2)

Equating (1) and (2)

1 - h= 1 + h

→ 2 h=0

→ h =0

Putting the value of h in either equation (1) or equation (2)

→ 1 = - 4 a k

→ $k=\frac{-1}{4a}$  

So, the equation of Parabola will be

$x^2=\pm 4 a (y+\frac{1}{4a})$

So, locus of Directrix will be:

$y\pm a=0$

Answer 2

Answer:It is given in my book.

Step-by-step explanation: